Radiating stars and Riccati equations in higher dimensions

Abstract

The objective of this study is to investigate spherically symmetric radiating stars undergoing gravitational collapse, in higher dimensional general relativity, inclusive of acceleration, expansion, shear, an electromagnetic field and a cosmological constant. Methods that can be used to obtain exact solutions to the boundary condition with/without a linear equation state are studied. Two distinct approaches are investigated. In the first approach, the boundary condition is expressed as a Riccati equation in terms of one of the dependent variables, and restrictions are placed to obtain new exact solutions. In the second approach, transformations that map the boundary condition into a new Riccati equation are investigated. The resulting new transformed equation is solved, by placing restrictions on the coefficients, to obtain new exact models. Special properties of the transformation are shown when appropriate restrictions on the parameters of the transformation are placed. This allows the order of the boundary condition to be reduced from a second order partial differential equation into a first order partial differential equation. The versatility of the transformation on other equations is exhibited when new solutions to the system of equations consisting of both the boundary condition and equation of state are obtained. When the dimension is set to four, some known solutions are recovered. It is shown that horizons can be identified by using a special case of the transformation. Our results elucidates the importance of the use of transformations that map the coordinates of differential equations into new and different coordinate systems.